Dynamical modeling of torso stability in running via hip–knee three pairs of six springs

Prior spring-mass locomotion models achieve stable gaits by prescribing a constant touchdown angle (TD angle) during the flight phase; however, they either exclude torso modeling or depend on online state feedback to stabilize the pitch angle of the torso. In contrast, evidence from biology and robotics suggests that coordinating monoarticular and biarticular hip-knee muscles supports whole-body stability with the simplified controller without online state feedback. However, this has only been verified through empirical and constructive approaches, rather than through dynamical modeling. To verify this hypothesis, we propose a new mathematical dynamical model, torso-hip-knee three pairs of six springs, a planar locomotion model that consists of a torso and coordinated springs imitating a three-pair six-muscle structure in the upper leg. The proposed dynamical model achieves stable running solely by giving a constant TD angle and a constant kicking angle relative to the torso, which control the dynamics during the flight and stance phases respectively. Numerical analysis utilizing Floquet multipliers demonstrates that self-stability emerges across stiffness parameters of coordinated springs. These results constitute the first mathematical evidence that muscle coordination, including biarticular muscles, can stabilize torso pitch during locomotion and provide guidelines for legged-robot design and rehabilitation assessment.